Some Commutativity Results
For Periodic Rings

Y. Madana Mohana Reddy; Dr. G. Shobhalatha; Dr. D.V. Rami Reddy

doi:https://doi.org/10.14445/22315373/IJMTT-V43P515

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 43 | Number 2 | Year 2017 | Article Id. IJMTT-V43P515 | DOI : https://doi.org/10.14445/22315373/IJMTT-V43P515

Some Commutativity Results For Periodic Rings

Y. Madana Mohana Reddy, Dr. G. Shobhalatha, Dr. D.V. Rami Reddy

Abstract

Herstein proved that a ring R which satisfies the identity , where n is a fixed positive integer greater than 1, must have nil commutator ideal. Abu-Khuzam provethat if n is a fixed positive integer greater than 1 and R is an n (n-1) – torsion free ring with unity such that for all x,y in R, then R is commutative. In Gupta proved that if R is a semi prime ring satisfying for all x,y in R, where Z(R) in the center of R, then R is commutative. In it is proved that a semi prime ring R such that for each x in R there exists a positive integer such that and for all y in r then R is commutative. In this direction we prove that if R is an n (n+1)-torsion free periodic ring such that and or and if the set of nilpotent elements of R is commutative. Then R is commutative. We know that R is periodic if for every x in R, there exist distinct positive integers m and n such that . By a theorem of Chancron , R is periodic if and only if for each x in R, there exists a positive integer k = k(x) and a polynomial with integer coefficients such that . Throughout this section R is an associative ring, Z(R) denotes the center of R and N denotes the set of nilpotent elements of R.

Keywords

center, periodic ring, direct sum, left and right ideals.

References

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[2] Abu-Khuzam.H A commutativity theorem for periodic rings, Math. Japonica.., 32No.1(1987) 1-3
[3] Bell H.E On some commutativity theorems of Herstein, Arch. Math.., 24(1973), 34-38
[4] Chacron, M. On a theorem of Herstein, Canad. J. Math.., 21(1969), 1348-1353
[5] Gupta, V. Some remarks on the commutativity of rings, Acta.Math.Acad.Sei.Hung, 36(1980), 232-236.
[6] Herstein, I.N. A generalization of a theorem of Jacobson, Amer. J.Math.., 73(1951), 756-762
[7] Herstein, I.N. Powermaps in rings, Michigan Math.J., 8(1960), 29-32
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Citation :

Y. Madana Mohana Reddy, Dr. G. Shobhalatha, Dr. D.V. Rami Reddy, "Some Commutativity Results For Periodic Rings," International Journal of Mathematics Trends and Technology (IJMTT), vol. 43, no. 2, pp. 93-97, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V43P515